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    Central Limit Theorem

         The central limit theorem (CLT) is one of the most useful theorems in statistics. It states that regardless of the population distribution, as long as all the variables are independently and identically distributed (i.i.d.), the sampling distribution of means will approach a normal distribution with mean equal to the population mean as the sampling size increases.

         Consider the population of Canada and imagine that you would like to find out what the mean height is. You start by taking a sample of size N and take the mean of that sample. But, this does not guarantee at all, that this will be the same as the population's mean. Now imagine that you continue to take more random samples of size N and find their means. Plotting all those means will give you a sampling distribution of means. The CLT states that this sampling distribution will approach a normal distribution as N grows large. Furthermore the mean of this sampling distribution will be approximately equal to the population mean as well. The sampling distribution variance will be approximately the population variance divided by N. The sampling standard deviation will be the square root of that. The central limit theorem evidently gives the ability to solve for the population mean with more accuracy than using one sample.  

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